Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Let be a generator of an exponentially stable operator semigroup in a Banach space, and let ？ be a linear bounded variable operator. Assuming that is sufficiently small in a certain sense for the equation , we derive exponential stability conditions. Besides, we do not require that for each , the “frozen” autonomous equation is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations. 1. Introduction and Statement of the Main Result In this paper, we investigate stability of linear nonautonomous equations in a Banach space, which can be considered as integrally small perturbations of autonomous equations. The stability theory of evolution equations in a Banach space is well developed, compare and confare with [1] and references therein, but the problem of stability analysis of evolution equations continues to attract the attention of many specialists despite its long history. It is still one of the most burning problems, because of the absence of its complete solution. One of the basic methods for the stability analysis is the direct Lyapunov method. By that method, many strong results were established, compare and confare with [2, 3]. But finding the Lyapunov functionals is usually a difficult mathematical problem. A fundamental approach to the stability of diffusion parabolic equations is the method of upper and lower solutions. A systematical treatment of that approach is given in [4]. In [5], stability conditions are established by a normalizing mapping. Note that a normalizing mapping enables us to use more complete information about the equation than a usual (number) norm. In [6], the “freezing” method for ordinary differential equations is extended to equations in a Banach space. About the recent results, see the interesting papers [7–11]. In particular, in [7] the Perron-Bellman theorem for evolutionary processes with exponential growth in Banach spaces is investigated. In the paper [8], a Rolewicz’s type theorem of in-solid function spaces is proved. Dragan and Morozan [9] established criteria for exponential stability of linear differential equations on ordered Banach spaces. Paper [10] deals with the stability and controllability of hyperbolic type abstract evolution equations. Pucci and Serrin [11] investigated the asymptotic stability for nonautonomous wave equations. Certainly, we could not survey the whole subject here and refer the reader to the previously listed publications and references given therein. Let be a complex Banach space with a